CISC 1100: Logic Fall 2014, X. Zhang, Fordham Univ. 1
Slide 2
Motivating example 2 Four machines A, B, C, D are connected on
a network. It is feared that a computer virus may have infected the
network. Your security team makes the following statements: If D is
infected, then so is C. If C is infected, then so is A. If D is
clean, then B is clean but C is infected. If A is infected, then
either B is infected or C is clean. Based on these statements, what
can you conclude about status of the four machines?
Slide 3
Smullyans Island Puzzle You meet two inhabitants of Smullyans
Island (where each one is either a liar or a truth-teller). A says,
Either B is lying or I am B says, A is lying Who is telling the
truth ? 3
Slide 4
Symbolic logic 4 Subjects: statements that is either true or
false, i.e., propositions Understand relations between statements
Equivalent statement: can we simplify and therefore understand a
statement better ? Contradictory statements: can both statements be
true ? Reasoning: does a statement follow from a set of hypothesis
? Application: solve logic puzzle, decide validity of
reasoning/proof
Proposition 6 Proposition: a statement which is either true or
false For example: Ten is less than seven. There are life forms on
other planets in the universe. A set of cardinality n has 2 n
subsets. The followings are not propositions: How are you ?
x+y
Binary => Decimal 58 Interpret binary numbers (transform to
base 10) 1101 = 1*2 3 +1*2 2 +0*2 1 +1*2 0 =8+4+0+1=13 Translate
the following binary number to decimal number 101011
Slide 59
Generally you can consider other bases 59 Base 8 (Octal number)
Use symbols: 0, 1, 2, 7 Convert octal number 725 to base 10: =7*8 2
+2*8 1 +5= Now you try: (1752) 8 = Base 16 (Hexadecimal) Use
symbols: 0, 1, 2, 9, A, B, C,D,E, F (10A) 16 = 1*16 2 +10*16 0
=..
Slide 60
Binary number arithmetic 60 Analogous to decimal number
arithmetics How would you perform addition? 0+0=0 0+1=1 1+1=10 (a
carry-over) Multiple digit addition: 11001+101= Subtraction: Basic
rule: Borrow one from next left digit
Slide 61
From Base 10 to Base 2: using table 61 Input : a decimal number
Output: the equivalent number in base 2 Procedure: Write a table as
follows 1. Find the largest twos power that is smaller than the
number 1. Decimal number 234 => largest twos power is 128 2.
Fill in 1 in corresponding digit, subtract 128 from the number
=> 106 3. Repeat 1-2, until the number is 0 4. Fill in empty
digits with 0 Result is 11101010 5122561286432168421 11101010
Slide 62
From Base 10 to Base 2: the recipe 62 Input : a decimal number
Output: the equivalent number in base 2 Procedure: 1. Divide the
decimal number by 2 2. Make the remainder the next digit to the
left of the answer 3. Replace the decimal number with the quotient
4. If quotient is not zero, Repeat 1-4; otherwise, done
Slide 63
Convert 100 to binary number 63 100 % 2 = 0 => last digit
100 / 2 = 50 50 % 2 = 0 => second last digit 50/2 = 25 25 % 2 =
1 => 3 rd last digit 25 / 2 = 12 12 % 2 = 0 => 4 th last
digit 12 / 2 = 6 6 % 2 = 0 => 5 th last digit 6 / 2 = 3 3 % 2 =
1 => 6 th last digit 3 / 2 =1 1 % 2 = 1 => 7 th last digit 1
/ 2 = 0 Stop as the decimal # becomes 0 The result is 1100100
Slide 64
Data Representation in Computer 64 In modern computers, all
information is represented using binary values. Each storage
location (cell): has two states low-voltage signal => 0
High-voltage signal => 1 i.e., it can store a binary digit,
i.e., bit Eight bits grouped together to form a byte Several bytes
grouped together to form a word Word length of a computer, e.g., 32
bits computer, 64 bits computer
Slide 65
Different types of data 65 Numbers Whole number, fractional
number, Text ASCII code, unicode Audio Image and graphics video How
can they all be represented as binary strings?
Slide 66
Representing Numbers 66 Positive whole numbers We already know
one way to represent them: i.e., just use base 2 number system All
integers, i.e., including negative integers Set aside a bit for
storing the sign 1 for +, 0 for Decimal numbers, e.g., 3.1415936,
100.34 Floating point representation: sign * mantissa * 2 exp 64
bits: one for sign, some for mantissa, some for exp.
Slide 67
Representing Text 67 Take English text for example Text is a
series of characters letters, punctuation marks, digits 0, 1, 9,
spaces, return (change a line), space, tab, How many bits do we
need to represent a character? 1 bit can be used to represent 2
different things 2 bit 2*2 = 2 2 different things n bit 2 n
different things In order to represent 100 diff. character Solve 2
n = 100 for n n =, here the refers to the ceiling of x, i.e., the
smallest integer that is larger than x:
Slide 68
There needs a standard way 68 ASCII code: American Standard
Code for Information Interchange ASCII codes represent text in
computers, communications equipment, and other devices that use
text.textcomputerscommunications 128 characters: 33 are
non-printing control characters (now mostly obsolete) [7] that
affect how text and space is processedcontrol characters [7] 94 are
printable characters space is considered an invisible graphic
space
Slide 69
ASCII code 69
Slide 70
There needs a standard way 70 Unicode
international/multilingual text character encoding system,
tentatively called Unicode Currently: 21 bits code space How many
diff. characters? Encoding forms: UTF-8: each Unicode character
represented as one to four 8-but bytes UTF-16: one or two 16-bit
code units UTF-32: a single 32-but code unit
Slide 71
How computer processing data? 71 Through manipulate digital
signals (high/low) Using addition as example 10000111 + 0001110
Input: the two operands, each consisting of 32 bits (i.e., 32
electronic signals) Output: the sum How ?
Slide 72
Digital Logic 72 Performs operation on one or more logic inputs
and produces a single logic output. Can be implemented
electronically using diodes or transistors Using electromagnetic
relays Or other: fluidics, optics, molecules, or even mechanical
elements We wont go into the physics of how is it done, instead we
focus on the input/output, and logic
Slide 73
Basic building block 73 Basic Digital logic is based on primary
functions (the basic gates): AND OR XOR NOT
Slide 74
AND Logic Symbol 74 Inputs Output If both inputs are 1, the
output is 1 If any input is 0, the output is 0
Slide 75
AND Logic Symbol 75 InputsOutput Determine the output Animated
Slide 0 0 0
Slide 76
AND Logic Symbol 76 InputsOutput Determine the output Animated
Slide 0 1 0
Slide 77
AND Logic Symbol 77 InputsOutput Determine the output Animated
Slide 1 1 1
Slide 78
AND Truth Table 78 To help understand the function of a digital
device, a Truth Table is used: InputOutput 000 010 100 111 AND
Function Every possible input combination
Slide 79
OR Logic Symbol 79 Inputs Output If any input is 1, the output
is 1 If all inputs are 0, the output is 0
Slide 80
OR Logic Symbol 80 InputsOutput Determine the output Animated
Slide 0 0 0
Slide 81
OR Logic Symbol 81 InputsOutput Determine the output Animated
Slide 0 1 1
Slide 82
OR Logic Symbol 82 InputsOutput Determine the output Animated
Slide 1 1 1
Slide 83
OR Truth Table 83 Truth Table InputOutput 000 011 101 111 OR
Function
Slide 84
The XOR function: if exactly one input is high, the output is
high If both inputs are high, or both are low, the output is low
XOR Gate 84
Slide 85
XOR Truth Table 85 Truth Table InputOutput 000 011 101 110 XOR
Function
Slide 86
NOT Logic Symbol 86 Input Output If the input is 1, the output
is 0 If the input is 0, the output is 1
Slide 87
NOT Logic Symbol 87 Input Output Determine the output Animated
Slide 0 1
Slide 88
NOT Logic Symbol 88 Input Output Determine the output Animated
Slide 1 0
Slide 89
Combinational logic 89 A circuit that utilizes more that one
logic function has Combinational Logic. How would your describe the
output of this combinational logic circuit?
Slide 90
Combinational Logic: Half Adder 90 Electronic circuit for
performing single digit binary addition Sum= A XOR B Carry = A AND
B ABSumCarry 0000 0110 1010 1101
Slide 91
Full Adder 91 One bit full adder with carry-in and carry-out
ABCI Q CO 00000 00110 01010 01101 10010 10101 11010 11111
Slide 92
Full adder: 4-digit binary addition 92 Chain 4 full-adders
together, lower digits carry-out is fed into the higher digit as
carry-in
Slide 93
Integrated Circuit 93 Also called chip A piece of silicon on
which multiple gates are embedded mounted on a package with pins,
each pin is either an input, input, power or ground Classification
based on # of gates VLSI (Very Large-Scale Integration) >
100,000 gates
Slide 94
CPU (Central Processing Unit) 94 Many pins to connect to
memory, I/O Instruction set: the set of machine instructions
supported by an architecture (such as Pentium) Move data (between
register and memory) Arithmetic Operations Logic Operations:
Floating point arithmetic Input/Output